A geometric framework for 3D leaf movement by orthonormal bases: a demonstration in Maranta leuconeura

Leaf orientation (posture) influences photosynthesis and plant responses to environmental cues. However, the existing methods for quantifying posture typically compress its inherently three-dimensional structure into scalar values or single vectors, leaving the 3D aspects of leaf movement poorly understood. For a more complete geometric description, we propose representing leaf-blade posture by an orthonormal basis (ONB), defined as three perpendicular unit vectors aligned with the developmental axes of the leaf. This ONB serves as a local coordinate system and corresponds to rotation matrices used to represent orientation in three-dimensional space, embedding leaf posture within the mathematical structure of the special orthogonal group SO(3). Using three-dimensional point-cloud data, we reconstructed an ONB aligned with the three axes of the leaf blade and quantified elevation and azimuth angles. When applied to diurnal posture changes, the resulting angular patterns were consistent with previous observations. We then visualized posture changes after gravitational perturbation as continuous rotational trajectories. These trajectories could be compared with mathematically defined geodesic shortest paths and used to simulate alternative reorientation routes that satisfy constraints. The rotational trajectories could also be separated into swing and twist components, which reflect distinct deformation modes. The ONB can be obtained not only from three-dimensional point clouds but also from other measurement tools, making the approach broadly applicable. Conceptually, ONB representation places leaf posture within the geometric structure of SO(3), enabling the use of well-established mathematical tools such as rotational distance and geodesics for analyzing leaf reorientation.